Let $M$ be a complex manifold of dimension $n$. Furthermore assume that we have a action of a Lie-Group $G$ on $M$ i.e. $G \times M \rightarrow M$, which is differential, meaning that for every $g \in G$ the map $g:M \rightarrow M$, $g:x \mapsto g \cdot x$ is differential (not holomorphic). Consider now a $(p,q)$-form $\alpha$ on $M$. My question is the following: Let $g \in G$ is it true that $g^{*}\overline{\partial} \alpha = \overline{\partial} g^{*}\alpha$ ? Or do we need that our group action is holomorphic? What is right?
bill
Let's try a simple example: $M=\mathbb C$, $\alpha=z$ (a $0$-form). Now take $g(z)=\bar z$ and observe that $g^* \bar \partial \alpha=0$ while $\bar \partial g^*\alpha=d\bar z$. In fact, $\bar \partial g^*\alpha=0$ exactly when $g$ is holomorphic.